This paper focuses on presenting an accurate, stable, efficient, and fast pseudospectral
method to solve tempered fractional differential equations (TFDEs) in both spatial and temporal
dimensions. We employ the Chebyshev interpolating polynomial for g at Gauss–Lobatto (GL) points
in the range [􀀀1, 1] and any identically shifted range. The proposed method carries with it a recast
of the TFDE into integration formulas to take advantage of the adaptation of the integral operators,
hence avoiding the ill-conditioning and reduction in the convergence rate of integer differential
operators. Via various tempered fractional differential applications, the present technique shows
many advantages; for instance, spectral accuracy, a much higher rate of running, fewer computational
hurdles and programming, calculating the tempered-derivative/integral of fractional order, and its
spectral accuracy in comparison with other competitive numerical schemes. The study includes
stability and convergence analyses and the elapsed times taken to construct the collocation matrices
and obtain the numerical solutions, as well as a numerical examination of the produced condition
number $k(A)% of the resulting linear systems. The accuracy and efficiency of the proposed method are
studied from the standpoint of the $L_2$ and $L_{\infty}$-norms error and the fast rate of spectral convergence.